BMW Sauber Ferrari from 2010 (C29), chassis theorically built by BMW+Sauber cooperation and using a Ferrari engine

Source: BMWblog

Was it really built by BMW or was it just a case of the team not renaming itself to Sauber until 2011 to avoid extra costs?

Yes it was almost 100% designed by BMW. However, once Peter bought his old team back from them he convened with his aerodynamicists to gauge where they were falling short the few years before and the suggestion was lack of air under the car. Peter quickly reached out to Ferrari and procured the engine and sought their input but all Ferrari told him was that in all likelihood their car lacked airflow under the chassis just like everyone else. The result was this hideous eye sore of a car which, I discussed with Bob Varsha and after he cued me in on some of these tidbits I told him I didn't think they had a shot with that thing because it looks all sorts of wrong to me. And while he didn't outright confirm my suspicions, he hinted that according to some investigative reporting by his compatriots at Speed Channel, I was definitely on the right track, but that the most important thing was that the team was safely grounded financially for at least 2 further seasons. Anything beyond that would come down to their performance over those 3 seasons. Boy was he ever right.

Caterham used Renault engines in their short time and their road car division (likely the worst and ugliest "exotic" cars ever peddled) are generally powered by Ford.

Though the manufacturer branding/badging was removed, the 2009 WCC winning car was for all intents and purposes 100% Honda dawning Mercedes power.

_________________Champions are made from something they have deep inside of them - a desire, a dream, a vision. They have last-minute stamina, they have to be a little faster, they have the skill & the will but the will must be stronger than the skill. Muhammad Ali